III. SYMBOLIC REASONING. (II.) (For I. see MIND, January, 1880.) BY HUGH MACCOLL. SYMBOLIC Logic (including Mathematics) may be defined as 'the science of reasoning by the aid of representative symbols ; these symbols being employed as synonymous substitutes for longer expressions that are required fre- quently '. The words in italics contain the pith and principle of the whole subject. When any expression, verbal or symbolic, of inconvenient length has to be written frequently in the course of an argument or investigation, we naturally cast about for some short and simple symbol to represent and replace it. This desire to economise time, space and labour is always, always has been, and always will be, the great motive power that sets going and keeps going the evolutionary progress of the science. What, for instance, was the primary object of the symbol x in such a case as 27365 x 7 ? Clearly to save the trouble of writing down an addition sum of seven rows of figures, each row being 27365. The same may be said of the symbol a 5 as a substitute for aaaaa, and of many others, including the remarkable and highly general symbol </>(&), which plays such an important part in the higher mathematics as a substitute for any expression whatever that contains x in any relation whatever as one of its constituents. As Symbolic Science advances and tackles more difficult pro- blems these conventional abbreviations afterwards combine among themselves and produce fresh expressions of incon- venient length and frequent recurrence which give birth in their turn to fresh representative substitutes to abbre- viations of abbreviations which in their powers of thought- condensation bear, on an average, the same ratio to the symbols they replace as these had done to their immediate progenitors. Thus it has been that the science of mathematics has slowly acquired its present marvellous
